Is the following assertion true?
"Let $X$ and $Y$ be two quasi-projective varieties and $f:X \longrightarrow Y$ a morphism which is injective and dominant. Then, the dimensions of $X$ and $Y$ coincide."
Certainly, if $f$ is just dominant, we have that $$\dim X \geq \dim Y,$$ as was shown in the answer to this question. So the problem is if equality holds if we impose injectivity.
If I am not wrong, we could deduce the result if $f$ were closed, by using the topolgical definition of dimension as in Hartshorne's book. However, since $X$ is not necessarily a projective variety but a quasi-projective, I guess we cannot use this argument.
Any help or suggestion is welcome.